Method of designing and manufacturing freeform surface off-axial imaging system

ABSTRACT

The present disclosure relates to a method of designing a freeform surface off-axial imaging system. The method comprises the steps of establishing an initial system and selecting feature fields; gradually enlarging a construction of feature field, and constructing the initial system into a freeform surface system; and expanding a construction area of each freeform surface of the freeform surface system, and reconstructing the freeform surface in an extended construction area.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims all benefits accruing under 35 U.S.C. § 119 fromChina Patent Application No. 201810034313.7, field on Jan. 12, 2018 inthe China National Intellectual Property Administration, disclosure ofwhich is incorporated herein by reference. The application is alsorelated to copending applications entitled, “FREEFORM SURFACE OFF-AXIALTHREE-MIRROR IMAGING SYSTEM”, filed on Dec. 23, 2018 Ser. No.16,231,535.

FIELD

The subject matter herein generally relates to a method of designingfreeform surface off-axial three-mirror imaging systems.

BACKGROUND

Compared with conventional rotationally symmetric surfaces, freeformsurfaces have asymmetric surfaces and more degrees of design freedom,which can reduce the aberrations and simplify the structure of thesystem. In recent years, freeform surfaces are often used in off-axialthree-mirror imaging system.

Conventional freeform surface off-axial three-mirror imaging systems aremostly obtained by direct design method, for example, a partialdifferential equation (PDE) design method, a simultaneous multiplesurface (SMS) method and a construction-iteration method (CI-3D).However, only a limited number of fields can be considered during adesign process of conventional methods. There are limitations ondesigning freeform surface off-axial three-mirror imaging systems withlarge relative apertures and large field.

BRIEF DESCRIPTION OF THE DRAWINGS

Implementations of the present technology will now be described, by wayof example only, with reference to the attached figures, wherein:

FIG. 1 is a flow diagram of an embodiment of a method of designing afreeform surface off-axial three-mirror imaging system.

FIG. 2 is a flow diagram of an embodiment of a method of designing afreeform surface off-axial three-mirror imaging system.

FIG. 3 is a method diagram of an embodiment of selecting feature fields.

FIG. 4 is an expanding direction diagram of an embodiment ofconstruction feature fields.

FIG. 5 is a flow diagram of an embodiment of expanding a constructionfeature field and constructing an initial system into a freeform surfacesystem.

FIG. 6 is a diagram of an embodiment of expanding the constructionfeature field and constructing the initial system into a freeformsurface system.

FIG. 7 is a flow diagram of an embodiment of extending a constructionarea of the freeform surface of the freeform surface system andreconstructing the freeform surface of the freeform surface system in anextended construction area.

FIG. 8 is a diagram of an embodiment of extending a construction areaand reconstructing freeform surfaces of the freeform surface system.

FIG. 9 is a diagram of an embodiment of a construction process of afreeform surface off-axis three-mirror imaging system.

FIG. 10 is a light path schematic view of an embodiment of a freeformsurface off-axis three-mirror imaging system.

FIG. 11 shows modulation transfer functions (MTF) of the freeformsurface off-axial three-mirror imaging system in FIG. 10.

FIG. 12 shows an average wave aberration of the freeform surfaceoff-axial three-mirror imaging system in FIG. 10.

FIG. 13 shows a relative distortion diagram of the freeform surfaceoff-axis three-mirror imaging system in FIG. 10.

DETAILED DESCRIPTION

The disclosure is illustrated by way of example and not by way oflimitation in the figures of the accompanying drawings in which likereferences indicate similar elements. It should be noted that referencesto “another,” “an,” or “one” embodiment in this disclosure are notnecessarily to the same embodiment, and such references mean “at leastone.”

It will be appreciated that for simplicity and clarity of illustration,where appropriate, reference numerals have been repeated among thedifferent figures to indicate corresponding or analogous elements. Inaddition, numerous specific details are set forth in order to provide athorough understanding of the embodiments described herein.

However, it will be understood by those of ordinary skill in the artthat the embodiments described herein can be practiced without thesespecific details. In other instances, methods, procedures and componentshave not been described in detail so as not to obscure the relatedrelevant feature being described. Also, the description is not to beconsidered as limiting the scope of the embodiments described herein.The drawings are not necessarily to scale and the proportions of certainparts have been exaggerated to better illustrate details and features ofthe present disclosure.

Several definitions that apply throughout this disclosure will now bepresented.

The term “contact” is defined as a direct and physical contact. The term“substantially” is defined to be that while essentially conforming tothe particular dimension, shape, or other feature that is described, thecomponent is not or need not be exactly conforming to the description.The term “comprising,” when utilized, means “including, but notnecessarily limited to”; it specifically indicates open-ended inclusionor membership in the so-described combination, group, series, and thelike.

FIG. 1 and FIG. 2 show one embodiment in relation to a method ofdesigning a freeform surface off-axial imaging system. The methodcomprises the following blocks:

block (B1), establishing an initial system and selecting a plurality offeature fields;

block (B2), selecting a feature field from the plurality of featurefields as a construction feature field, gradually expanding theconstruction feature field and constructing the initial system into afreeform surface system comprising a freeform surface; and

block (B3), extending a construction area of the freeform surface of thefreeform surface system to obtain an extended construction area, andreconstructing the freeform surface of the freeform surface system inthe extended construction area, to obtain the freeform surface off-axialthree mirror imaging system.

In block (B1), the initial system comprises a plurality of initialsurfaces, and each of the plurality of initial surfaces corresponds toone freeform surface of the freeform surface off-axial imaging systems.The plurality of initial surfaces can be planar surfaces or sphericalsurfaces. The locations of the plurality of initial surfaces can beselected according to the actual needs of the freeform surface off-axialimaging systems. The number of the plurality of initial surfaces can beselected according to the actual needs. In one embodiment, the initialsystem is an initial planar three-mirror imaging system, the initialplanar three-mirror imaging system comprises three initial planar.

The number of the plurality of feature fields is much larger than thenumber of the initial surfaces of the initial system. In one embodiment,the number of the plurality of feature fields is more than 50 times ofthe number of the initial surfaces. In one embodiment, the number of theplurality of feature fields is 50 to 60 times of the number of theinitial surfaces. The feature fields can be selected according to actualneeds. In one embodiment, M feature fields are isometric sampled insagittal direction and meridian direction, the M feature fields aredefined as φ_(i) (i=1, 2, . . . M), and (0, φ_(y0)) is defined as acentral field. In one embodiment, the initial system is the initialplanar three-mirror imaging system, a field of the initial planarthree-mirror imaging system in meridian direction is from about 8° toabout 9°, and a field of the initial planar three-mirror imaging systemin sagittal direction is from about −30° to about 30°. Referring to FIG.2, the initial planar three-mirror imaging system is symmetric about aYOZ plane, thus, only half of the field in the sagittal direction can beconsidered. A feature field is sampled at intervals of 0.5°, threefeature fields are sampled in the sagittal direction, and 61 featurefields are sampled in the meridian direction, therefore, 183 featurefields are selected.

K feature rays are selected in each of the M feature fields. A method ofselecting the K feature rays comprises steps of: an aperture of each ofthe M fields is divided into N equal parts; and, P feature rays atdifferent aperture positions in each of the N equal parts are selected.As such, K=M×N×P different feature rays correspond to different aperturepositions and different fields are selected. The aperture can be circle,rectangle, square, oval or other shapes. In one embodiment, the apertureof each of the M fields is a circle, and a circular aperture of each ofthe M fields is divided into N angles with equal interval φ, as such,N=2π/φ; then, P different aperture positions are fixed along a radialdirection of each of the N angles. Therefore, K=M×N×P different featurerays correspond to different aperture positions and different fields areselected. In one embodiment, the aperture of each field is divided into14 equal parts; and, 7 feature rays at different aperture positions ineach of the 14 equal parts are selected. As such, 98 different featurerays are selected in each field.

In block (B1), the freeform surface off-axial three-mirror imagingsystem is usually symmetric in the sagittal direction, thus, whenconstructing the initial system into the freeform surface system, onlythe feature rays of a half field in the sagittal direction can beconsidered. Referring to FIG. 3, in one embodiment, the central field(0, φ_(y0)) is as a first construction feature field φ₁, duringexpanding the construction feature field, the construction feature fieldin the meridional direction is expanded in both positive and negativedirections, the construction feature field in the sagittal direction isexpanded in positive direction. In one embodiment, the initial system isthe initial planar three-mirror imaging system, a field of the initialplanar three-mirror imaging system in meridian direction is from about8° to about 9°, the field range is only about 1°, thus, during expandingthe construction feature field, the construction feature field isexpanded only in the sagittal direction, and the three feature fields inthe meridional direction are considered simultaneously in each step ofconstruction.

Referring to FIG. 4 and FIG. 5, the initial system comprises L initialsurfaces, and the L initial surfaces are defined as L_(j)(j=1, 2, . . .L). Referring to FIG. 5, block (B2) comprises the steps of:

block (B21), selecting at least one field of the M feature fields φ_(i)(i=1, 2, . . . M) as the first construction feature field, andconstructing an initial surface L₁ into a freeform surface N₁ in thefirst construction feature field;

block (B22), adding at least one feature field to expand the firstconstruction feature field to obtain a second construction featurefield, and constructing an initial surface L₂ into a freeform surface N₂in the second construction feature field; and

block (B23), in the same way as step (S22), expanding the constructionfeature field by adding at least one feature field each time to obtainedan expanded construction feature field, and constructing an initialsurface L_(i)(j=3, L) into a freeform surface N_(i)(j=3, L) in theexpanded construction feature field, until the L initial surfaces areconstructed into L freeform surfaces.

In block (B21), a method of constructing the initial surface L₁ into thefreeform surface N₁ in the first construction feature field comprises:calculating a plurality of feature data points P_(i) (i=1, 2 . . . K) onthe freeform surface N_(i); and surface fitting the plurality of featuredata points to obtain an equation of the freeform surface N₁.

A surface Ω is defined as the freeform surface N₁, a surface Ω′ isdefined as a surface located adjacent to and before the surface Ω, and asurface Ω″ is defined as a surface located adjacent to and after thesurface Ω. The intersections of the plurality of feature rays R_(i)(i=1, 2 . . . K) with the surface Ω are defined as the first featuredata points P_(i) (i=1, 2 . . . K). The plurality of feature rays R_(i)(i=1, 2 . . . K) are intersected with the surface Ω′ at a plurality ofstart points S_(i) (i=1, 2 . . . K), and intersected with the surface Ω″at a plurality of end points E_(i) (i=1, 2 . . . K). When the surface Qand the plurality of feature rays R_(i) (i=1, 2 . . . K) are determined,the plurality of start points S_(i) (i=1, 2 . . . K) of the feature raysR_(i) (i=1, 2 . . . K) can also be determined. The plurality of endpoints E_(i) (i=1, 2 . . . K) can also be obtained based on theobject-image relationship or given mapping relationship. Under idealconditions, the feature rays R_(i) (i=1, 2 . . . K) emitted from theplurality of start points S_(i) (i=1, 2 . . . K) on the surface Ω′; passthrough the first feature data points P_(i) (i=1, 2 . . . K) on thesurface Ω; intersect with the surface Ω″ at the plurality of end pointsE_(i) (i=1, 2 . . . K); and finally intersect with the image plane atthe plurality of ideal target points T_(i,ideal) (i=1, 2 . . . K). Ifthe surface Ω″ is the target plane, the plurality of end points E_(i),(i=1, 2 . . . K) are the plurality of ideal target points I_(i) (i=1, 2. . . K). If there are other surfaces between the surface Q and thetarget plane, the plurality of end points E_(i) (i=1, 2 . . . K) are thepoints on the surface a″, which make the first variation of the opticalpath length between the first feature data points P_(i) (i=1, 2 . . . K)and their corresponding target points zero. δS=δ∫_(r) _(i) ^(T) ^(i)nds=0, wherein ds is the differential elements of the optical pathlength along the plurality of feature rays R_(i) (i=1, 2 . . . K), ndenotes the refractive index of the medium, and δ denotes a differentialvariation.

A method of calculating the plurality of feature data points P_(i) (i=1,2 . . . K) comprises:

block (a): defining a first intersection of a first feature light ray R₁and the initial surface as a feature data point P₁;

block (b): when an ith (1≤i≤K−1) feature data point P_(i) (1≤i≤K−1) hasbeen obtained, a unit normal vector {right arrow over (N)}_(i) at theith (1≤i≤K−1) feature data point P_(i) (1≤i≤K−1) can be calculated basedon the vector form of Snell's law;

block (c): making a first tangent plane through the ith (1≤i≤K−1)feature data point P_(i), (1≤i≤K−1), and (K−i) second intersections canbe obtained by the first tangent plane intersecting with remaining (K−i)feature rays; a second intersection Q_(i+1), which is nearest to the ith(1≤i≤K−1) feature data point P_(i) (1≤i≤K−1), is fixed; and a featureray corresponding to the second intersection Q_(i+1) is defined asR_(i+1), a shortest distance between the second intersection Q_(i+1) andthe ith feature data point P_(i) (1≤i≤K−1) is defined as d_(i);

block (d): making a second tangent plane at (i−1) feature data pointsthat are obtained before the ith feature data point P_(i) (1≤i≤K−1)respectively; thus, (i−1) second tangent planes can be obtained, and(i−1) third intersections can be obtained by the (i−1) second tangentplanes intersecting with a feature ray R_(i+1); in each of the (i−1)second tangent planes, each of the (i−1) third intersections and itscorresponding feature data point form an intersection pair; theintersection pair, which has the shortest distance between a thirdintersection and its corresponding feature data point, is fixed; and thethird intersection and the shortest distance is defined as Q′_(i+1) andd′_(i) respectively;

block (e): comparing d_(i) and d′_(i), if d_(i)≤d′_(i), Q_(i+1) is takenas the next feature data point P_(i+1) (1≤i≤K−1); otherwise, Q′_(i+1) istaken as the next feature data point P_(i+1) (1≤i≤K−1); and

block (f): repeating steps from (b′) to (e′), until the plurality offeature data points P_(i) (i=1, 2 . . . K) are all calculated.

In block (b), the unit normal vector {right arrow over (N)}_(i)(1≤i≤K−1) at each of the feature data point P_(i) (1≤i≤K−1) can becalculated based on the vector form of Snell's Law. When the freeformsurface N₁ is a refractive surface,

$\begin{matrix}{{\overset{arrow}{N}}_{i} = \frac{{n\;\overset{\primearrow\prime}{r_{i}}} - {n\;\overset{arrow}{r_{i}}}}{{{n\;\overset{\primearrow\prime}{r_{i}}} - {n\;\overset{arrow}{r_{i}}}}}} & (1)\end{matrix}$

$\;{\overset{arrow}{r_{i}} = \frac{\overset{harpoonup}{P_{i}S_{i}}}{\overset{harpoonup}{P_{i}S_{i}}}}$is a unit vector along a direction of an incident ray for the freeformsurface N₁;

${\overset{arrow}{r}}_{i}^{\prime} = \frac{\overset{arrow}{E_{i}P_{i}}}{\overset{arrow}{E_{i}P_{i}}}$is a unit vector along a direction for an exit ray of the first freeformsurface; and n, n′ is refractive index of a media before and after thefreeform surface N₁ respectively.

Similarly, when the freeform surface N₁ is a reflective surface,

$\begin{matrix}{{\overset{arrow}{N}}_{i} = \frac{{\overset{arrow}{r}}_{i}^{\prime} - {\overset{arrow}{r}}_{i}}{{{\overset{arrow}{r}}_{i}^{\prime} - {\overset{arrow}{r}}_{i}}}} & (2)\end{matrix}$

The unit normal vector {right arrow over (N)}_(i) at each of theplurality of feature data points P_(i) (i=1, 2 . . . K) is perpendicularto the first tangent plane at each of the plurality of feature datapoints P_(i) (i=1, 2 . . . K). Thus, the first tangent plane at each ofthe plurality of feature data points P_(i) (i=1, 2 . . . K) can beobtained.

In one embodiment, the space of the initial system is defined as a firstthree-dimensional rectangular coordinates system. The propagationdirection of beams is defined as a Z-axis, and the Z-axis isperpendicular to an XOY plane.

A method of surface fitting the plurality of feature data points P_(i)(i=1, 2 . . . K) comprises:

block (B211): surface fitting the plurality of feature data points P_(i)(i=1, 2 . . . K) to a sphere in the first three-dimensional rectangularcoordinates system, and obtaining a curvature c of the sphere and thecenter of curvature (x_(c), y_(c), z_(c)) corresponding to the curvaturec of the sphere;

block (B212): defining the feature data point (x_(o), y_(o), z_(o))corresponding to a chief ray of the central field angle among the entirefield-of-view (FOV) as the vertex of the sphere, defining a secondthree-dimensional rectangular coordinates system by the vertex of thesphere as origin and a line passing through the center of curvature andthe vertex of the sphere as a Z′-axis;

block (B213): transforming the coordinates (x_(i), y_(i), z_(i)) and thenormal vector (α_(i), β_(i), γ_(i)), of the plurality of feature datapoints P_(i) (i=1, 2 . . . K) in the first three-dimensional rectangularcoordinates system, into the coordinates (x′_(i), y′_(i), z′_(i)) andthe normal vector (α′_(i), β′_(i), γ′_(i)), of the plurality of featuredata points P_(i) (i=1, 2 . . . K) in the second three-dimensionalrectangular coordinates system;

block (B214): fitting the plurality of feature data points P_(i), (i=1,2 . . . K) into a conic surface equation of a conic surface in thesecond three-dimensional rectangular coordinates system, based on thecoordinates (x′_(i), y′_(i), z′_(i)) and the curvature c of the sphere,and obtaining the conic constant k; and

block (B215): removing the coordinates and the normal vector of theplurality of first feature data points P_(i) (i=1, 2 . . . K), on theconic surface in the second three-dimensional rectangular coordinatessystem, from the coordinates (x′_(i), y′_(i), z′_(i)) and the normalvector (α′_(i), β′_(i), γ′_(i)), to obtain a residual coordinate and aresidual normal vector; and fitting the residual coordinate and theresidual normal vector to obtain a polynomial surface equation; theequation of the freeform surface N₁ can be obtained by adding the conicsurface equation and the polynomial surface equation.

Generally, the imaging systems are symmetric about the YOZ plane.Therefore, the tilt angle θ of the sphere, in the Y′O′Z′ plane of thesecond three-dimensional rectangular coordinates system relative to inthe YOZ plane of the first three-dimensional rectangular coordinatessystem, is:

$\theta = {{\arctan( \frac{y_{o} - y_{c}}{z_{o} - z_{c}} )}.}$

The relationship between the coordinates (x′_(i), y′_(i), z′_(i)) andthe coordinates (x_(i), y_(i), z_(i)) of each of the plurality offeature data points P_(i) (i=1, 2 . . . K) can be expressed asfollowing:

$\{ {\begin{matrix}{{x_{i}^{\prime} = {x_{i} - x_{o}}}\mspace{265mu}} \\{y_{i}^{\prime} = {{( {y_{i} - y_{o}} )\mspace{14mu}\cos\mspace{14mu}\theta} - {( {z_{i} - z_{o}} )\mspace{14mu}\sin\mspace{14mu}\theta}}} \\{z_{i}^{\prime} = {{( {y_{i} - y_{o}} )\mspace{14mu}\sin\mspace{14mu}\theta} + {( {z_{i} - z_{o}} )\mspace{14mu}\cos\mspace{14mu}\theta}}}\end{matrix}.} $

The relationship between the normal vector (α′_(i), β′₁, γ′_(i)) and thenormal vector (α_(i), β_(i), γ_(i)) of each of the plurality of featuredata points P_(i) (i=1, 2 . . . K) can be expressed as following:

$\{ {\begin{matrix}{{\alpha_{i}^{\prime} = \alpha_{i}}} \\{\beta_{i}^{\prime} = {{\beta_{i}\mspace{14mu}\cos\mspace{14mu}\theta} - {\gamma_{i}\mspace{14mu}\sin\mspace{14mu}\theta}}} \\{\gamma_{i}^{\prime} = {{\beta_{i}\mspace{14mu}\sin\mspace{14mu}\theta} + {\gamma_{i}\mspace{14mu}\cos\mspace{14mu}\theta}}}\end{matrix}.} $

In the second three-dimensional rectangular coordinates system, thecoordinates and the normal vector of the plurality of feature datapoints P_(i) (i=1, 2 . . . K) on the conic surface are defined as(x′_(i), y′_(i), z′_(is)) and (α′_(is), β′_(is), y′_(is)) respectively.The Z′-axis component of the normal vector is normalized to −1. Theresidual coordinate (x″_(i), y″_(i), z″_(i)) and the residual normalvector (α″_(i), β″_(i), −1) can be obtained, wherein, (x_(i)″, y_(i)″,z_(i)″)=(x_(i)′, y_(i)′, z_(i)′−z_(is)′) and

$( {\alpha_{i}^{''},\beta_{i}^{''},{- 1}} ) = {( {{{- \frac{\alpha_{i}^{\prime}}{\gamma_{i}^{\prime}}} + \frac{\alpha_{is}^{\prime}}{\gamma_{is}^{\prime}}},{{- \frac{\beta_{i}^{\prime}}{\gamma_{i}^{\prime}}} + \frac{\beta_{is}^{\prime}}{\gamma_{is}^{\prime}}},{- 1}} ).}$

In block (B215), a method of surface fitting the residual coordinate andthe residual normal vector comprises:

in the second three-dimensional rectangular coordinates system,expressing a polynomial surface of the freeform surface off-axialthree-mirror imaging system by the polynomial surface equation leavingout the conic surface term, the polynomial surface can be expressed interms of the following equation:

${z = {{f( {x,{y;P}} )} = {\sum\limits_{j = 1}^{J}\;{P_{j}{g_{j}( {x,y} )}}}}},$wherein g_(j)(x,y) is one item of the polynomial, and P=(p₁, p₂, . . . ,p_(J))^(T) is the coefficient sets;

acquiring a first sum of squares d₁(P), of residual coordinatedifferences in z′ direction between the residual coordinate value(x″_(i), y″_(i), z″_(i)) (i=1, 2, . . . , K) and the freeform surface;and a second sum of squares d₂(P), of modulus of vector differencesbetween the residual normal vector N_(i)=(α″_(i), β″_(i), −1) (i=1, 2, .. . , K) and a normal vector of the freeform surface, wherein the firstsum of squares d₁(P) is expressed in terms of a first equation:

${{d_{1}(P)} = {{\sum\limits_{i = 1}^{I}\;\lbrack {z_{i} - {f( {x_{i}^{''},{y_{i}^{''};P}} )}} \rbrack^{2}} = {( {Z - {A_{1}P}} )^{T}( {Z - {A_{1}P}} )}}},$and

the second sum of squares d₂(P) is expressed in terms of a secondequation:

${d_{2}\mspace{11mu}(P)} = {{\sum\limits_{i = 1}^{I}\{ {\lbrack {u_{i} - {f_{x^{''}}( {x_{i}^{''},{y_{i}^{''};P}} )}} \rbrack^{2} + \lbrack {v_{i} - {f_{y^{''}}( {x_{i}^{''},{y_{i}^{''};P}} )}} \rbrack^{2}} \}} = {{( {U - {A_{2}P}} )^{T}( {U - {A_{2}P}} )} + {( {V - {A_{3}P}} )^{T}( {V - {A_{3}P}} )}}}$

wherein, Z=(z₁, z₂, . . . , z₁)^(T), U=(u₁, u₂, . . . , u₁)^(T), V=(v₁,v₂, . . . , v₁)^(T),

${A_{1} = \begin{pmatrix}{g_{1}( {x_{1}^{''},y_{1}^{''}} )} & {g_{2}( {x_{1}^{''},y_{1}^{''}} )} & \cdots & {g_{J}( {x_{1}^{''},y_{1}^{''}} )} \\{g_{1}( {x_{2}^{''},y_{2}^{''}} )} & {g_{2}( {x_{2}^{''},y_{2}^{''}} )} & \cdots & {g_{J}( {x_{2}^{''},y_{2}^{''}} )} \\\vdots & \vdots & \; & \vdots \\{g_{1}( {x_{I}^{''},y_{I}^{''}} )} & {g_{2}( {x_{I}^{''},y_{I}^{''}} )} & \cdots & {g_{J}( {x_{I}^{''},y_{I}^{''}} )}\end{pmatrix}},{A_{2} = \begin{pmatrix}{g_{1}^{x}( {x_{1}^{''},y_{1}^{''}} )} & {g_{2}^{x}( {x_{1}^{''},y_{1}^{''}} )} & \cdots & {g_{J}^{x}( {x_{1}^{''},y_{1}^{''}} )} \\{g_{1}^{x}( {x_{2}^{''},y_{2}^{''}} )} & {g_{2}^{x}( {x_{2}^{''},y_{2}^{''}} )} & \cdots & {g_{J}^{x}( {x_{2}^{''},y_{2}^{''}} )} \\\vdots & \vdots & \; & \vdots \\{g_{1}^{x}( {x_{I}^{''},y_{I}^{''}} )} & {g_{2}^{x}( {x_{I}^{''},y_{I}^{''}} )} & \cdots & {g_{J}^{x}( {x_{I}^{''},y_{I}^{''}} )}\end{pmatrix}},{{A_{3} = \begin{pmatrix}{g_{1}^{y}( {x_{1}^{''},y_{1}^{''}} )} & {g_{2}^{y}( {x_{1}^{''},y_{1}^{''}} )} & \cdots & {g_{J}^{y}( {x_{1}^{''},y_{1}^{''}} )} \\{g_{1}^{y}( {x_{2}^{''},y_{2}^{''}} )} & {g_{2}^{y}( {x_{2}^{''},y_{2}^{''}} )} & \cdots & {g_{J}^{y}( {x_{2}^{''},y_{2}^{''}} )} \\\vdots & \vdots & \; & \vdots \\{g_{1}^{y}( {x_{I}^{''},y_{I}^{''}} )} & {g_{2}^{y}( {x_{I}^{''},y_{I}^{''}} )} & \cdots & {g_{J}^{y}( {x_{I}^{''},y_{I}^{''}} )}\end{pmatrix}};}$

obtaining an evaluation function,P=(A ^(T) ₁ A ₁ +wA ₂ ^(T) A ₂ +wA ₃ ^(T) A ₃)⁻¹·(A ₁ ^(T) Z+wA ₂ ^(T)U+wA ₃ ^(T) V),

wherein w is a weighting greater than 0;

selecting different weightings w and setting a gradient ∇ƒ(P) of theevaluation function equal to 0, to obtain a plurality of differentvalues of P and a plurality of freeform surface shapes z=f (x, y; P)corresponding to each of the plurality of different values of P; and

choosing a final freeform surface shape Ω_(opt) which has a best imagingquality from the plurality of freeform surface shapes z=f(x, y; P).

The methods of constructing other initial surfaces of the initial systeminto the freeform surfaces are substantially the same as the method ofconstructing the freeform surface N₁, except that the constructionfeature fields are different. Methods of calculating the feature datapoints on other freeform surfaces are the same as the method ofcalculating the feature data points on freeform surface N₁. Methods ofsurface fitting the plurality of feature data points on other freeformsurfaces are the same as the method of fitting the plurality of featuredata points on the freeform surface N₁.

In one embodiment, in block (B21), selecting one field of the M featurefields φ_(i) (i=1, 2, . . . M) as the first construction feature field;and in block (B22) and block (B23), expanding the construction featurefield by adding one feature field each time to obtained the expandedconstruction feature field.

Referring to FIGS. 6 and 7, block (B3) comprises:

block (B31), adding at least one feature field to extend the expandedconstruction feature field in block (B23) to obtain a constructionfeature field “I”, and constructing the freeform surface N₁ into afreeform surface N′₁ in the construction feature field “I”;

block (B32), adding at least one feature field to extend theconstruction feature field “I” to obtain a construction feature field“II”, and constructing the freeform surface N₂ into a freeform surfaceN′₂ in the construction feature field “II”;

block (B33), in the same way as block (B32), extending the constructionfeature field by adding at least one feature field each time to obtainedan extended construction feature field, and constructing L freeformsurfaces in the freeform surface system into L new freeform surfaces;and repeating this step until the M feature fields are used up; and

block (B34), if there are at least one feature field of the M featurefields is not used on at least one freeform surface, and reconstructingthe at least one freeform surface using the M feature fields.

Referring to FIG. 8, in one embodiment, when the freeform surface N_(k)in the freeform surface system is reconstructed into a freeform surfaceN′_(k), the M feature fields are used up; and other freeform surfacesexcept the freeform surface N′_(k) are reconstructed using all the Mfeature fields.

In block (B31), the intersections of the feature light rays in theconstruction feature field “I” and the freeform surface N₁ can beobtained by Snell's law and object image relationship. The intersectionsare the feature data points on the freeform surface N′₁. Solving anormal vector at each feature data point based on the object imagerelationship, and surface fitting the plurality of feature data pointsto obtain an equation of the freeform surface N′₁. A method of surfacefitting the plurality of feature data points to obtain the equation ofthe freeform surface N′₁ is the same as the method of fitting theplurality of feature data points P_(i)(i=1, 2 . . . K) to obtain theequation of the freeform surface N₁ in block (B2).

In one embodiment, extending the construction feature field by addingone feature field each time to obtained the extended constructionfeature field. Block (B3) comprises: block (B31), adding a feature fieldφ_(L+1), and constructing the freeform surface N₁ into a freeformsurface N′₁ in the feature fields (φ₁, φ₁ . . . φ_(L+1)); block (B32),adding a feature field φ_(L+2), and constructing the freeform surface N₂into a freeform surface N′₂ in the feature fields (φ₁, φ₁ . . .φ_(L+2)); and block (B32), in the same way as block (B32), extending theconstruction feature field by adding one feature field each time toobtained an extended construction feature field, and constructing Lfreeform surfaces in the freeform surface system into L new freeformsurfaces, until the M feature fields are used up.

Furthermore, a step of optimizing the freeform surface off-axial imagingsystem obtained in block (B3) by using the freeform surface off-axialimaging systems as an initial system of optimization can be performed.

In one embodiment, after block (B3), further comprises enlarge thefreeform surface off-axis imaging system by a certain multiple.

In one embodiment, after the freeform surface off-axial three-mirrorimaging system is designed, further manufacturing the freeform surfaceoff-axial three-mirror imaging system.

Referring to FIG. 9, in one embodiment, constructing a tertiary mirrorin the feature field φ₁=0.5°; adding the construction feature field andconstructing a secondary mirror and a primary mirror in turn; after thetertiary mirror, the secondary mirror, and the primary mirror areconstructed, a freeform surface extension process is performed, and areconstruction sequence of the extension process is the tertiary mirror,the secondary mirror, and the primary mirror in turn.

Freeform surface off-axial imaging systems with large relative apertureand wide field can be obtained by the above method of designing thefreeform surface off-axial three-mirror imaging system. Furthermore, anaberration compensation for the constructed feature field can beperformed in the process of expanding the construction feature field,and thus an image quality of the freeform surface off-axial three-mirrorimaging system is improved.

Referring to FIG. 10, a freeform surface off-axial three-mirror imagingsystem 100 is designed with the above method to prove the above method.The freeform surface off-axial three-mirror imaging system 100 comprisesa primary mirror 102, a secondary mirror 104, and a tertiary mirror 106adjacent and spaced from each other. The secondary mirror 104 is used asa stop surface. A surface shape of each of the primary mirror 102, thesecondary mirror 104, and the tertiary mirror 106 is a freeform surface.The feature rays exiting from the light source would be successivelyreflected by the primary mirror 102, the secondary mirror 104 and thetertiary mirror 106 to form an image on an image sensor 108.

A first three-dimensional rectangular coordinates system (X,Y,Z) isdefined by a location of the primary mirror 102; a secondthree-dimensional rectangular coordinates system (X′,Y′,Z′) is definedby a location of the secondary mirror 104; and a third three-dimensionalrectangular coordinates system (X″,Y″,Z″) is defined by a location ofthe tertiary mirror 106.

A vertex of the primary mirror 102 is an origin of the firstthree-dimensional rectangular coordinates system (X,Y,Z). A horizontalline passing through the vertex of the primary mirror 102 is defined asa Z-axis; in the Z-axis, to the left is negative, and to the right ispositive. A Y-axis is substantially perpendicular to the Z-axis and in aplane shown in FIG. 8; in the Y-axis, to the upward is positive, and tothe downward is negative. An X-axis is substantially perpendicular to aYZ plane; in the X-axis, to the inside is positive, and to the outsideis negative.

A reflective surface of the primary mirror 102 in the firstthree-dimensional rectangular coordinates system (X,Y,Z) is an xypolynomial freeform surface; and an xy polynomial equation can beexpressed as follows:

${z( {x,y} )} = {\frac{c( {x^{2} + y^{2}} )}{1 + \sqrt{1 - {( {1 + k} ){c^{2}( {x^{2} + y^{2}} )}}}} + {\sum\limits_{i = 1}^{N}\;{A_{i}x^{m}{y^{n}.}}}}$

In the xy polynomial equation, z represents surface sag, c representssurface curvature, k represents conic constant, while A_(i) representsthe ith term coefficient. Since the freeform surface off-axialthree-mirror imaging system 100 is symmetrical about a YOZ plane, soeven order terms of x can be only remained. At the same time, higherorder terms will increase the fabrication difficulty of the off-axialthree-mirror optical system with freeform surfaces 100. In oneembodiment, the reflective surface of the primary mirror 102 is aneighth-order polynomial freeform surface of xy without odd items of x;and an equation of the eighth-order polynomial freeform surface of xycan be expressed as follows:

${z( {x,y} )} = {\frac{c( {x^{2} + y^{2}} )}{1 + \sqrt{1 - {( {1 + k} ){c^{2}( {x^{2} + y^{2}} )}}}} + {A_{2}y} + {A_{3}x^{2}} + {A_{5}y^{2}} + {A_{7}x^{2}y} + {A_{9}y^{3}} + {A_{10}x^{4}} + {A_{12}x^{2}y^{2}} + {A_{14}y^{4}} + {A_{16}x^{4}y} + {A_{18}x^{2}y^{3}} + {A_{20}y^{5}} + {A_{21}x^{6}} + {A_{23}x^{4}y^{2}} + {A_{25}x^{2}y^{4}} + {A_{27}y^{6}} + {A_{29}x^{6}y} + {A_{31}x^{4}y^{3}} + {A_{33}x^{2}y^{5}} + {A_{35}y^{7}} + {A_{36}x^{8}} + {A_{38}x^{6}y^{2}} + {A_{40}x^{4}y^{4}} + {A_{42}x^{2}{y^{6}.}}}$

In one embodiment, the values of c, k, and A_(i) in the equation of theeighth-order polynomial freeform surface of xy of the reflective surfaceof the primary mirror 102 are listed in TABLE 1. However, the values ofc, k, and A_(i) in the eighth order xy polynomial equation are notlimited to TABLE 1.

TABLE 1 c   2.0905292558E−03 k −1.21892397257173 A₂ −7.4132168689E−01 A₃−1.0489399277E−03 A₅ −1.0932187535E−03 A₇   1.8886774078E−07 A₉  3.9367755612E−08 A₁₀   1.8215989925E−10 A₁₂   2.0074956485E−10 A₁₄  3.4948173329E−09 A₁₆ −1.5663370553E−13 A₁₈ −2.3270934790E−13 A₂₀−4.1540232365E−11 A₂₁ −7.5480347775E−17 A₂₃   4.3918989072E−16 A₂₅−8.2899003141E−15 A₂₇   1.7435762237E−13 A₂₉   1.3339713122E−19 A₃₁  3.0368239305E−18 A₃₃   8.0061751667E−18 A₃₅ −4.7950132667E−17 A₃₆  5.1808677261E−23 A₄₀ −1.2808456638E−20 A₄₂   6.5955002284E−19

A vertex of the secondary mirror 104 is an origin of the secondthree-dimensional rectangular coordinates system (X′,Y′,Z′). The secondthree-dimensional rectangular coordinates system (X′,Y′,Z′) is obtainedby moving the first three-dimensional rectangular coordinates system(X,Y,Z) along a Z-axis negative direction and a Y-axis negativedirection. In one embodiment, The second three-dimensional rectangularcoordinates system (X′,Y′,Z′) is obtained by moving the firstthree-dimensional rectangular coordinates system (X,Y,Z) for about272.306 mm along a Y-axis negative direction, and then moving for about518.025 mm along a Z-axis negative direction, and then rotating alongthe counterclockwise direction for about 31.253° with the X axis as therotation axis. A distance between the origin of the firstthree-dimensional rectangular coordinates system (X,Y,Z) and the originof the second three-dimensional rectangular coordinates system(X′,Y′,Z′) is about 585.235 mm.

In the second three-dimensional rectangular coordinates system(X′,Y′,Z′), a reflective surface of the secondary mirror is an x′y′polynomial freeform surface. An x′y′ polynomial surface equation can beexpressed as follows:

${z^{\prime}( {x^{\prime},y^{\prime}} )} = {\frac{c^{\prime}( {x^{\prime 2} + y^{\prime 2}} )}{1 + \sqrt{1 - {( {1 + k^{\prime}} ){c^{\prime 2}( {x^{\prime 2} + y^{\prime 2}} )}}}} + {\sum\limits_{i = 1}^{N}\;{A_{i}^{\prime}x^{\prime\; m}y^{\prime\; n}}}}$

In the x′y′ polynomial freeform surface equation, z′ represents surfacesag, c′ represents surface curvature, k′ represents conic constant,while A_(i)′ represents the ith term coefficient. Since the freeformsurface off-axial three-mirror imaging system 100 is symmetrical aboutY′Z′ plane, so even-order terms of x′ can be only remained. At the sametime, higher order terms will increase the fabrication difficulty of thefreeform surface off-axial three-mirror imaging system 100. In oneembodiment, the reflective surface of the secondary mirror is aneighth-order polynomial freeform surface of x′y′ without odd items ofx′. An equation of the eighth-order polynomial freeform surface of x′y′can be expressed as follows:

${z^{\prime}( {x^{\prime},y^{\prime}} )} = {\frac{c^{\prime}( {x^{\prime 2} + y^{\prime 2}} )}{1 + \sqrt{1 - {( {1 + k^{\prime}} ){c^{\prime 2}( {x^{\prime 2} + y^{\prime 2}} )}}}} + {A_{2}^{\prime}y^{\prime}} + {A_{3}^{\prime}x^{\prime 2}} + {A_{5}^{\prime}y^{\prime 2}} + {A_{7}^{\prime}x^{\prime 2}y^{\prime}} + {A_{9}^{\prime}y^{\prime 3}} + {A_{10}^{\prime}x^{\prime 4}} + {A_{12}^{\prime}x^{\prime 2}y^{\prime 2}} + {A_{14}^{\prime}y^{\prime 4}} + {A_{16}^{\prime}x^{\prime 4}y^{\prime}} + {A_{18}^{\prime}x^{\prime 2}y^{\prime 3}} + {A_{20}^{\prime}y^{\prime 5}} + {A_{21}^{\prime}x^{\prime 6}} + {A_{23}^{\prime}x^{\prime 4}y^{\prime 2}} + {A_{25}^{\prime}x^{2\prime}y^{\prime 4}} + {A_{27}^{\prime}y^{\prime 6}} + {A_{29}^{\prime}x^{\prime 6}y^{\prime}} + {A_{31}^{\prime}x^{\prime 4}y^{\prime 3}} + {A_{33}^{\prime}x^{\prime 2}y^{\prime 5}} + {A_{35}^{\prime}y^{\prime 7}} + {A_{36}^{\prime}x^{\prime 8}} + {A_{38}^{\prime}x^{\prime 6}y^{\prime 2}} + {A_{40}^{\prime}x^{\prime 4}y^{\prime 4}} + {A_{42}^{\prime}x^{\prime 2}y^{\prime 6}}}$

In one embodiment, the values of c′, k′, and A_(i)′ in the equation ofthe eighth-order polynomial freeform surface of x′y′ are listed in TABLE2. However, the values of c′, k′, and A_(i)′ in the equation of theeighth-order polynomial freeform surface of x′y′ are not limited toTABLE 2.

TABLE 2 c′   1.2419254309E−03 k′ 11.4470143760146 A₂′   2.6039347329E−01A₃′ −1.5571495307E−03 A₅′ −1.8499719195E−03 A₇′   3.1123110562E−07 A₉′−7.2750566448E−08 A₁₀′ −1.0565839855E−08 A₁₂′ −2.4835983560E−08 A₁₄′−1.6584532798E−08 A₁₆′ −2.1393403767E−12 A₁₈′ −1.6810005234E−12 A₂₀′−1.3947835440E−11 A₂₁′ −3.1895103407E−14 A₂₃′ −8.4530987667E−13 A₂₅′−4.1377307296E−13 A₂₇′   8.5456614986E−14 A₂₉′   9.6174103349E−15 A₃₁′−7.1427426063E−15 A₃₃′ −3.1310574623E−15 A₃₅′   2.8784215520E−15 A₃₆′−6.8582114820E−17 A₄₀′ −1.5699572247E−15 A₄₂′   6.5300451569E−17

A vertex of the tertiary mirror 106 is an origin of the thirdthree-dimensional rectangular coordinates system (X″,Y″,Z″). The thirdthree-dimensional rectangular coordinates system (X″,Y″,Z″) is obtainedby moving the second three-dimensional rectangular coordinates system(X′,Y′,Z′) along a Z′-axis positive direction and a Y′-axis positivedirection. In one embodiment, the third three-dimensional rectangularcoordinates system (X″,Y″,Z″) is obtained by moving the secondthree-dimensional rectangular coordinates system (X′,Y′,Z′) for about346.467 mm along a Z′-axis negative direction, and then moving for about141.540 mm along a Y′-axis negative direction, and then rotating alongthe counterclockwise direction for about 20.079° with the X′-axis s asthe rotation axis. A distance between the origin of the thirdthree-dimensional rectangular coordinates system (X″,Y″,Z″) and theorigin of the second three-dimensional rectangular coordinates system(X′,Y′,Z′) is about 374.263 mm.

In the third three-dimensional rectangular coordinates system(X′,Y′,Z′), a reflective surface of the tertiary mirror 106 is an x″y″polynomial freeform surface. An x″y″ polynomial surface equation can beexpressed as follows:

${z^{''}( {x^{''},y^{''}} )} = {\frac{c^{''}( {x^{''2} + y^{''2}} )}{1 + \sqrt{1 - {( {1 + k^{''}} ){c^{''2}( {x^{''2} + y^{''2}} )}}}} + {\sum\limits_{i = 1}^{N}\;{A_{i}^{''}x^{''\; m}{y^{''\; n}.}}}}$

In the x″y″ polynomial freeform surface equation, z″ represents surfacesag, c″ represents surface curvature, k″ represents conic constant,while A_(i)″ represents the ith term coefficient. Since the freeformsurface off-axial three-mirror imaging system 100 is symmetrical aboutY″Z″ plane, so even-order terms of x″ can be only remained. At the sametime, higher order terms will increase the fabrication difficulty of thefreeform surface off-axial three-mirror imaging system 100. In oneembodiment, the reflective surface of the tertiary mirror 106 is aneighth-order polynomial freeform surface of x″y″ without odd items ofx″. An equation of the eighth-order polynomial freeform surface of x″y″can be expressed as follows:

${z^{''}( {x^{''},y^{''}} )} = {\frac{c^{''}( {x^{''2} + y^{''2}} )}{1 + \sqrt{1 - {( {1 + k^{''}} ){c^{''2}( {x^{''2} + y^{''2}} )}}}} + {A_{2}^{''}y^{''}} + {A_{3}^{''}x^{''2}} + {A_{5}^{''}y^{''2}} + {A_{7}^{''}x^{''2}y^{''}} + {A_{9}^{''}y^{''3}} + {A_{10}^{''}x^{''4}} + {A_{12}^{''}x^{''2}y^{''2}} + {A_{14}^{''}y^{''4}} + {A_{16}^{''}x^{''4}y^{''}} + {A_{18}^{''}x^{''2}y^{''3}} + {A_{20}^{''}y^{''5}} + {A_{21}^{''}x^{''6}} + {A_{23}^{''}x^{''4}y^{''2}} + {A_{25}^{''}x^{2''}y^{''4}} + {A_{27}^{''}y^{''6}} + {A_{29}^{''}x^{''6}y^{''}} + {A_{31}^{''}x^{''4}y^{''3}} + {A_{33}^{''}x^{''2}y^{''5}} + {A_{35}^{''}y^{''7}} + {A_{36}^{''}x^{''8}} + {A_{38}^{''}x^{''6}y^{''2}} + {A_{40}^{''}x^{''4}y^{''4}} + {A_{42}^{''}x^{''2}{y^{''6}.}}}$

In one embodiment, the values of c″, k″, and A_(i)″ in the eighth orderx″y″ polynomial surface equation are listed in TABLE 3. However, thevalues of c″, k″, and A_(i)″ in the eighth order x″y″ polynomial surfaceequation are not limited to TABLE 3.

TABLE 3 c″ −2.3117990132E−03 k″ −9.0353982823E−02 A₂″   2.3151773248E−02A₃″   1.3411119803E−04 A₅″   9.7954301258E−05 A₇″   4.0553954355E−08 A₉″  2.1746460869E−08 A₁₀″   1.6438448222E−10 A₁₂″   2.2410934162E−10 A₁₄″  4.5429699280E−11 A₁₆″   1.1872420633E−13 A₁₈″   1.5798764234E−13 A₂₀″  1.3425352595E−14 A₂₁″   3.1961386864E−16 A₂₃″   8.2640583723E−16 A₂₅″  5.0212511084E−17 A₂₇″   1.5492827151E−16 A₂₉″   7.1429537548E−19 A₃₁″  1.1507858347E−18 A₃₃″ −6.5780484704E−19 A₃₅″   4.2158929949E−18 A₃₆″  7.6045895800E−22 A₄₀″ −1.0675040865E−22 A₄₂″   2.0087308226E−20

The materials of the primary mirror 102, the secondary mirror 104 andthe tertiary mirror 106 can be aluminum, beryllium or other metals. Thematerials of the primary mirror 102, the secondary mirror 104 and thetertiary mirror 106 can also be silicon carbide, quartz or otherinorganic materials. A reflection enhancing coating can also be coatedon the metals or inorganic materials to enhance the reflectivityperformance of the three mirrors. In one embodiment, the reflectionenhancing coating is a gold film. A size of each of the primary mirror102, the secondary mirror 104 and the tertiary mirror 106 can bedesigned according to actual needs.

In the first three-dimensional rectangular coordinates system (X,Y,Z), adistance along the Z-axis negative direction between a center of theimage sensor 108 and the vertex of the tertiary mirror 106 is about391.895 mm. The center of the image sensor 108 deviates from the Z axisin the positive direction of the Y axis, and a deviation is about 33.491mm. An angle of the image sensor 108 with the XY plane in the clockwisedirection is about 7.435°. A size of the image sensor 108 can beselected according to actual needs. In one embodiment, the size of theimage sensor 108 is 5 μm×5 μm.

An effective entrance pupil diameter of the freeform surface off-axialthree-mirror imaging system 100 is about 60 mm.

The freeform surface off-axial three-mirror imaging system 100 adopts anoff-axis field in meridian direction. A field angle of the freeformsurface off-axial three-mirror imaging system 100 is larger than orequal to 60°×1°. In one embodiment, the field angle of the freeformsurface off-axial three-mirror imaging system 100 is about 60°×1°, anangle along the sagittal direction is from about −30° to about 30°, andan angle along the meridian direction is from about 8° to about −9°.

A wavelength of the freeform surface off-axial three-mirror imagingsystem 100 is not limited, in one embodiment, the wavelength of thefreeform surface off-axial three-mirror imaging system 100 is from about380 nm to about 780 nm.

An effective focal length (EFL) of the freeform surface off-axialthree-mirror imaging system 100 is about 150 mm.

A F-number of the freeform surface off-axial three-mirror imaging system100 is less than or equal to 2.5. A relative aperture (D/f) is thereciprocal of the F-number. In one embodiment, the F-number is 2.5, andthe relative aperture (D/f) is 0.4.

Referring to FIG. 11, a modulation transfer functions (MTF) of thefreeform surface off-axial three-mirror imaging system 100 in visibleband of partial field angles are close to the diffraction limit. Itshows that an imaging quality of the freeform surface off-axialthree-mirror imaging system 100 is high.

Referring to FIG. 12, an average wave aberration of the freeform surfaceoff-axial three-mirror imaging system 100 is 0.089λ, λ=530.5 nm. Itshows that an imaging quality of the freeform surface off-axialthree-mirror imaging system 100 is high.

Referring to FIG. 13, it can be seen that a maximum relative distortionof the freeform surface off-axial three-mirror imaging system 100 is8.1%. The freeform surface off-axial three-mirror imaging system 100 hassmall relative distortion compared to other free-form off-axisthree-reverse imaging systems with large field angles. It shows that animaging error is small and the image quality is high.

Since the freeform surface off-axial three-mirror imaging system 100 hasno center block, the freeform surface off-axial three-mirror imagingsystem 100 can achieve a large field angle and a large imaging range,the field angle is about 60°×1°. The F-number of the freeform surfaceoff-axial three-mirror imaging system 100 is 2.5, high resolution imagescan be obtained by the freeform surface off-axial three-mirror imagingsystem 100. A structure of the freeform surface off-axial three-mirrorimaging system 100 is compact.

The applications of the freeform surface off-axial three-mirror imagingsystem 100 comprises earth observation, space target detection,astronomical observations, Multi-spectral thermal imaging, anddimensional mapping. The freeform surface off-axial three-mirror imagingsystem 100 can be used in the visible band or the infrared band.

Depending on the embodiment, certain blocks/steps of the methodsdescribed may be removed, others may be added, and the sequence ofblocks may be altered. It is also to be understood that the descriptionand the claims drawn to a method may comprise some indication inreference to certain blocks/steps. However, the indication used is onlyto be viewed for identification purposes and not as a suggestion as toan order for the blocks/steps.

The embodiments shown and described above are only examples. Even thoughnumerous characteristics and advantages of the present technology havebeen set forth in the foregoing description, together with details ofthe structure and function of the present disclosure, the disclosure isillustrative only, and changes may be made in the detail, especially inmatters of shape, size, and arrangement of the parts within theprinciples of the present disclosure, up to and including the fullextent established by the broad general meaning of the terms used in theclaims. It will therefore be appreciated that the embodiments describedabove may be modified within the scope of the claims.

What is claimed is:
 1. A method of manufacturing a freeform surfaceoff-axial imaging system comprising: step(S1): establishing an initialsystem comprising a plurality of initial surfaces, wherein the pluralityof initial surfaces are defined as L₁, L₂, . . . and L_(j), j representsa number of the plurality of initial surfaces, and j is an integergreater than or equal to 3, and each of the plurality of initialsurfaces L₁, L₂, . . . and L_(j) corresponds to one freeform surface ofthe freeform surface off-axial imaging system; and selecting a pluralityof feature fields, wherein a number of the plurality of feature fieldsis defined as M, and a value of M is larger than a value of j; step(S2):selecting at least one field from the plurality of feature fields as afirst construction feature field, and constructing an initial surface L₁into a first freeform surface N₁ in the first construction featurefield; adding at least one feature field from the plurality of featurefields to the first construction feature field to expand the firstconstruction feature field to obtain a second construction featurefield, and constructing an initial surface L₂ into a second freeformsurface N₂ in the second construction feature field; and the rest isdeduced from this, until a last initial surface L_(j) is constructedinto a jth freeform surface N_(j) in a jth construction feature field;step(S3): adding at least one feature field from the plurality offeature fields to the jth construction feature field to extend the jthconstruction feature field to obtain a j+1th construction feature field,and constructing the first freeform surface N₁ into a first new freeformsurface N′₁ in the j+1th construction feature field; adding at least onefeature field from the plurality of feature fields to the j+1thconstruction feature field to extend the j+1th construction featurefield to obtain a j+2th construction feature field, and constructing thesecond freeform surface N₂ into a second new freeform surface N′₂ in thej+2th construction feature field; and the rest is deduced from this,until the jth freeform surface N_(j) is constructed to a jth newfreeform surfaces N′j, and the rest is deduced from this, until theplurality of feature fields are used up; and after the plurality offeature fields are used up, there are at least one feature field of theplurality of feature fields is not used on at least one new freeformsurface of the new freeform surfaces N′₁, N′₂, . . . , and N′_(j),reconstructing the at least one new freeform surface using the pluralityof feature fields, to obtain all freeform surfaces of the freeformsurface off-axial imaging system, and therefore to obtain the freeformsurface off-axial imaging system; and step(S4): manufacturing a freeformsurface off-axial imaging system comprising a plurality of mirrors withfreeform surface based on a structure of the freeform surface off-axialimaging system obtained in step (S3).
 2. The method of claim 1, whereinthe value of M is more than 50 times of the value of L.
 3. The method ofclaim 2, wherein the value of M ranges from about 50 to about 60 timesof the value of j.
 4. The method of claim 1, wherein the plurality offeature fields are isometrically sampled in a sagittal direction and ameridian direction.
 5. The method of claim 4, wherein the plurality offeature fields are isometrically sampled at intervals of 0.5 degree. 6.The method of claim 5, wherein three feature fields are sampled in thesagittal direction, and 61 feature fields are sampled in the meridiandirection.
 7. The method of claim 4, wherein during expanding theconstruction feature field, the construction feature field in themeridional direction is expanded in both a positive direction and anegative direction, and the construction feature field in the sagittaldirection is expanded in the positive direction.
 8. The method of claim1, wherein the initial system is an initial planar three-mirror imagingsystem.
 9. The method of claim 8, wherein a field of the initial planarthree-mirror imaging system in a meridian direction is from 8 degrees to9 degrees, and a field of the initial planar three-mirror imaging systemin sagittal direction is from −30 degrees to 30 degrees.
 10. The methodof claim 1, wherein a method of constructing the initial surface L₁ intothe first freeform surface in the first construction feature fieldcomprises: calculating a plurality of feature data points P_(i), whereini=1, 2 . . . or K, on the first freeform surface; and surface fittingthe plurality of feature data points P_(i) to obtain an equation of thefirst freeform surface.
 11. The method of claim 10, wherein a method ofcalculating the plurality of feature data points P_(i), wherein i=1, 2 .. . or K, comprises: step (a): defining a first intersection of a firstfeature light ray R₁ and the initial surface as a feature data point P₁;step (b): when an ith feature data point P_(i) has been obtained, a unitnormal vector {right arrow over (N)}_(t) at the ith feature data pointP_(i) is calculated based on a vector form of Snell's law, wherein instep (b), 1≤i≤K−1; step (c): making a first tangent plane through theith feature data point P_(i) in step (b), and K−i second intersectionsare obtained by the first tangent plane intersecting with remaining K−ifeature rays; a second intersection Q_(i+1), which is nearest to the ithfeature data point P_(i), is fixed; and a feature ray corresponding tothe second intersection Q_(i+1) is defined as R_(i+1), a shortestdistance between the second intersection Q_(i+i) and the ith featuredata point P_(i) is defined as d₁; step (d): making a second tangentplane at i−1 feature data points that are obtained before the ithfeature data point P_(i) respectively; thus, i−1 second tangent planescan be obtained, and i−1 third intersections can be obtained by the i−1second tangent planes intersecting with a feature ray R_(i+1); in eachof the i−1 second tangent planes, each of the i−1 third intersectionsand its corresponding feature data point form an intersection pair; theintersection pair, which has the shortest distance between a thirdintersection and a corresponding feature data point of the thirdintersection, is fixed; and the third intersection and the shortestdistance is defined as Q′_(i+1) and d′_(i) respectively; step (e):comparing d_(i) and d′_(i), if d_(i)≤d′_(i), Q_(i+1) is taken as thenext feature data point P_(i), wherein 1≤i≤K−1; otherwise, is taken asthe next feature data point P_(i+1) wherein 1≤i≤K−1; and step (f):repeating steps from (b′) to (e′), until the plurality of feature datapoints P_(i), wherein i=1, 2 . . . or K, are all calculated.
 12. Themethod of claim 1, wherein selecting one field of the plurality offeature fields as the first construction feature field, and expandingthe first construction feature field by adding one feature field toobtain the second construction feature field.
 13. The method of claim 1,wherein a plurality of intersections of a plurality of feature lightrays in the j+1th construction feature field and the first freeformsurface N₁ are obtained by Snell's law and object image relationship,the plurality of intersections are a plurality of feature data points onthe first new freeform surface N′₁; and surface fitting the plurality offeature data points to obtain an equation of the first new freeformsurface N′₁.
 14. The method of claim 1, further comprising a step ofoptimizing the freeform surface off-axial imaging system by using thefreeform surface off-axial imaging system as an initial system ofoptimization.
 15. The method of claim 1, further comprising a step ofenlarging the freeform surface off-axis imaging system.